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# Universe, Geometry of

For centuries, mathematicians and physicists believed that the universe was accurately described by the axioms of Euclidean geometry, which now form a standard part of high school mathematics. Some of the distinctive properties of Euclidean geometry follow:

• Straight lines go on forever.
• Parallel lines are unique. That is, given a line and a point not on that line, there is one and only one parallel line that passes through the given point.
• Parallel lines are equidistant. That is, two points moving along parallel lines at the same speed will maintain a constant distance from each other.
• The angles of a triangle add to 180°.
• The circumference of a circle is proportional to its radius, r, (C = 2πr ) and the area, A, of a circle is proportional to the square of the radius (A = πr 2), where pi (π) is defined to be approximately 3.14 .

The last four properties are characteristic of a "flat" space that has no intrinsic curvature. (See figure below.)

Other types of geometry, called non-Euclidean geometries, violate some or all of the properties of Euclidean geometry. For example, on the surface of a sphere (a positively curved space), the closest thing to a straight path is a great circle, or the path you would follow if you walked straight forward without turning right or left. But great circles do not go on forever: They loop around once and then close up.

If "lines" are interpreted to mean "great circles," the other properties of Euclidean geometry are also false in spherical geometry. Parallel great circles do not exist at all. The positive curvature causes great circles that start out in the same direction to converge. For example, the meridians on a globe converge at the north and south poles. The angles of a spherical triangle add up to more than 180°, and the circumference and area of spherical circles are smaller than 2 πr and πr 2, respectively. (See figure on p. 120.)

In the early years of the nineteenth century, three mathematicians Karl Friedrich Gauss, Janos Bolyai, and Nikolai Lobachevskiindependently discovered non-Euclidean geometries with negative curvature. In these geometries, lines that start out in the same direction tend to diverge from each other. The angles of a triangle add to less than 180°. The circumference of a circle grows exponentially faster than the radius. Although negatively curved spaces are difficult to visualize and depict because we are accustomed to looking at the world through "Euclidean eyes," you can get some idea by looking at the figure directly below.

## Curved Space and General Relativity

For a long time curved geometries remained in the realm of pure mathematics. However, in 1915 Albert Einstein proposed the theory of general relativity, which stated that our universe is actually curved, at least in some places. Any large amount of matter tends to impart a positive curvature to the space around it. The force that we call gravity is a result of the curvature of space.*

*An oft-repeated aphorism is: "Matter tells space how to curve, and space tells matter how to move."

In general relativity, rays of light play the role of lines. Because space is positively curved near the Sun, Einstein predicted that rays of light from a distant star that just grazed the surface of the Sun would bend towards the Sun. (See figure below.) To observers on Earth, such stars would appear displaced from where they really were. Most of the time, of course, the light from the Sun completely obscures any light that comes from distant stars, so we cannot see this effect. But Einstein's prediction was dramatically confirmed by observations made during the total solar eclipse of 1919. Such "gravitational lensing" effects are now commonplace in astronomy.

Einstein, like many later scientists, was also very interested in the overall geometry of the universe. The stars are, after all, aberrations in a universe that is mostly empty space. If you distributed all the matter in the universe evenly, would the curvature be positive, negative, or zero? Interestingly, on this issue Einstein was not a revolutionary. He assumed, in his first model of the universe, that the overall geometry would be flat. It was a Russian physicist, Alexander Friedmann,* who first pointed out in 1922 that Einstein's equations could be solved just as well by positively or negatively curved universes.

*Alexander Alexandrovitch Friedmann (18881925) was a founder of dynamic meteorology.

Friedmann also showed that the curvature was not just a mathematical curiosity: The fate of the entire universe depended on it. This dependence occurs because the curvature of the universe is closely related to the density of matter. Namely, if there is less than the critical density of matter in the universe, then the universe is negatively curved, but if there is more than the critical density of matter, then it is positively curved. If the universe is negatively curved, it keeps expanding forever because the gravitational attraction caused by the small amount of matter is not enough to rein in the universe's expansion. But if the universe is positively curved, the universe ends with a "Big Crunch" because the gravitational attraction causes the expansion of space to slow down, stop, and reverse. If the amount of matter is "just right," then the curvature is neither positive nor negative, and space is Euclidean. In a flat universe, the growth of space slows down but never quite stops.

## Big Bang Theory

Both of Friedmann's models of the universe assume that it began with a bang. Although Einstein was skeptical of these theories at first, the Big Bang theory received powerful support in 1929, when Edwin Hubble showed that distant galaxies are receding from Earth at a rate proportional to their distance. For the first time, it was possible to determine the age of the universe. (Hubble's original estimate was 2 billion years; the currently accepted range is 10 to 15 billion years.)

Another piece of evidence for the Big Bang theory came in 1964, in the form of the cosmic microwave background, a sort of afterglow of the Big Bang. The discovery of this radiation, which comes from all directions of the sky, confirmed that the early universe (about 300,000 years after the Big Bang) was very hot, and also very uniform. Every part of the cosmic fireball was at almost exactly the same temperature.

After 1964, the Big Bang theory became a kind of scientific orthodoxy. But few people realized that it was really several theories, not onethe question of the curvature and ultimate fate of the universe was still wide open.

Over the 1960s and 1970s, a few physicists began to have misgivings about the Friedmann models. One centered on the assumption of homogeneity, which Friedmann (and Einstein) had made as a mathematical convenience. Though the microwave data had confirmed this nicely, there still was no good explanation for it. And, in fact, a careful look at the night sky seems to show the distribution of mass in the universe is not very uniform. Our immediate neighborhood is dominated by the Milky Way. The Milky Way is itself part of the Local Group of galaxies, which in turn lies on the fringes of the Virgo Cluster, whose center is 60 million light-years away. Astronomers have found evidence for even larger structures, such as a "Great Wall" of galaxies 300 million light-years across. How could a universe that started out uniform and featureless develop such big clumps?

A second problem was the "flatness problem." In cosmology, the branch of physics that deals with the beginnings (and endings) of the universe, the density of matter in the universe is denoted by Omega (abbreviated O or Ω). By convention, a flat (Euclidean) universe has a density of 1, and this is called the critical density. In more ordinary language, the critical density translates to about one hydrogen atom per cubic foot, or one poppy seed spread out over the volume of the Earth. It is an incredibly low density.

The trouble is that Omega does not stay constant unless it equals 1 or 0. If the universe started out with a density a little less than 1, then Omega would rapidly decrease until it was barely greater than 0. If it started out with a density of 1, then Omega would stay at 1. It is very difficult to create a universe where, after 10 billion years, Omega is still partway between 0 and 1. And yet that is what the observational data suggested: The most careful estimates to date put the density of ordinary matter, plus the density of dark matter that does not emit enough light to be seen through telescopes, at about 0.30. This seemed to require a very "fine-tuned" universe. Physicists view with suspicion any theory that suggests our universe looks the way it does because of a very specific, or lucky, choice of constants. They would rather deduce the story of our universe from general principles.

## Revising the Big Bang Theory

In 1980, Alan Guth proposed a revised version of the Big Bang that has become the dominant theory, because it answers both the uniformity and flatness paradoxes. And unlike Friedmann's models, it makes a clear prediction about the geometry of the universe: The curvature is 0 (or very close to it), and the geometry is Euclidean.

Guth's theory, called inflation, actually came out of particle physics, not cosmology. At an unimaginably small time after the Big Bangroughly a trillionth of a trillionth of a nanosecondthe fundamental forces that bind atoms together went through a "phase transition" that made them repel rather than attract. For just an instant, a billionth of a trillionth of a nanosecond, the universe flew apart at a phenomenal, faster-than-light rate. Then, just as suddenly, the inflationary epoch was over. The universe reached another phase transition, the direction of the nuclear forces reversed, and space resumed growing at the normal pace predicted by the ordinary Big Bang models.

Inflation solves the uniformity problem because everything we can see today came from an incredibly tiny, and therefore homogeneous, region of the pre-inflationary universe. And it solves the flatness problem because, like a balloon being stretched taut, the universe lost any curvature that it originally had. The prediction that space is flat has become one of the most important tests for inflation.

Until recently, the inflationary model has not been consistent with astronomical observations. Its supporters had to argue that the observed clumps of galaxies, up to hundreds of millions of light-years in size, can be reconciled with a very homogeneous early universe. They also had to believe that something is out there to make up the difference between the observed Omega of 0.3 and the predicted Omega of 1.

A series of new measurements in 1999 and 2000, however, provided some real victories for inflation. First, they showed that the microwave background does have some slight variations, and the size of these variations is consistent with a flat universe. More surprisingly, the data suggest that most of the "missing matter" is not matter but energy, in an utterly mysterious form called dark energy or vacuum energy. In essence, this theory claims that there is something about the vacuum itself that pushes space apart. The extra energy causes space to become flat, but it also makes space expand faster and faster (rather than slower and slower, as in Friedmann's model).

Until these measurements, the theory of vacuum energy was not accepted. It had originally been proposed by Einstein, who later abandoned it and called it his greatest mistake. Yet years later, cosmologists said that the vacuum energy was the most accurately known constant in physics, to 120 decimal places, and composes 70 percent of the matter-energy in our universe.

There is no way to tell at this time whether vacuum energy is the final word or whether it will be discarded again. It is clear, though, that the geometry of space will continue to play a crucial role in understanding the most distant past and the most distant future of our universe.

see also Cosmos; Einstein, Albert; Euclid and His Contributions; Geometry, Spherical; Solar System Geometry, History of; Solar System Geometry, Modern Understandings of.

Dana Mackenzie

## Bibliography

Lightman, Alan, and Roberta Brawer. Origins: The Lives and Worlds of Modern Cosmologists. Cambridge, MA: Harvard University Press, 1990.

Steele, Diana. "Unveiling the Flat Universe." Astronomy (August) 2000: 4650.

Weeks, Jeffrey. The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds. New York: Marcel Dekker, 1985.